Spatial Interpolation: A Brief Introduction

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Spatial Interpolation A Brief Introduction

• Eugene Brusilovskiy

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General Outline

• Introduction to interpolation
• Deterministic interpolation methods
• Some basic statistical concepts
• Autocorrelation and First Law of Geography
• Geostatistical Interpolation
• Introduction to variography
• Kriging models

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What is Interpolation?

• Assume we are dealing with a variable which has
  meaningful values at every point within a region
  (e.g., temperature, elevation, concentration of
  some mineral). Then, given the values of that
  variable at a set of sample points, we can use an
  interpolation method to predict values of this
  variable at every point
• For any unknown point, we take some form of
  weighted average of the values at surrounding
  points to predict the value at the point where
  the value is unknown
• In other words, we create a continuous surface
  from a set of points
• As an example used throughout this presentation,
  imagine we have data on the concentration of gold
  in western Pennsylvania at a set of 200 sample
  locations

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Appropriateness of Interpolation

• Interpolation should not be used when there isnt
  a meaningful value of the variable at every point
  in space (within the region of interest)
• That is, when points represent merely the
  presence of events (e.g., crime), people, or some
  physical phenomenon (e.g., volcanoes, buildings),
  interpolation does not make sense.
• Whereas interpolation tries to predict the value
  of your variable of interest at each point,
  density analysis (available, for instance, in
  ArcGISs Spatial Analyst) takes known quantities
  of some phenomena and spreads it across the
  landscape based on the quantity that is measured
  at each location and the spatial relationship of
  the locations of the measured quantities.
• Source http//webhelp.esri.com/arcgisdesktop/9.2/
  index.cfm?TopicNameUnderstanding_density_analysis
  

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Interpolation vs. Extrapolation

• Interpolation is prediction within the range of
  our data
• E.g., having temperature values for a bunch of
  locations all throughout PA, predict the
  temperature values at all other locations within
  PA
• Note that the methods we are talking about are
  strictly those of interpolation, and not
  extrapolation
• Extrapolation is prediction outside the range of
  our data
• E.g., having temperature values for a bunch of
  locations throughout PA, predict the temperature
  values in Kazakhstan

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First Law of Geography

• Everything is related to everything else, but
  near things are more related than distant
  things.
• Waldo Tobler (1970)
• This is the basic premise
  behind interpolation, and
  near points generally
  receive
  higher weights
  than far away points

Reference TOBLER, W. R. (1970). "A computer
movie simulating urban growth in the Detroit
region". Economic Geography, 46(2) 234-240.
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Methods of Interpolation

• Deterministic methods
• Use mathematical functions to calculate the
  values at unknown locations based either on the
  degree of similarity (e.g. IDW) or the degree of
  smoothing (e.g. RBF) in relation with neighboring
  data points.
• Examples include
• Inverse Distance Weighted (IDW)
• Radial Basis Functions (RBF)
• Geostatistical methods
• Use both mathematical and statistical methods to
  predict values at all locations within region of
  interest and to provide probabilistic estimates
  of the quality of the interpolation based on the
  spatial autocorrelation among data points.
• Include a deterministic component and errors
  (uncertainty of prediction)
• Examples include
• Kriging
• Co-Kriging

Reference http//www.crwr.utexas.edu/gis/gishydro
04/Introduction/TermProjects/Peralvo.pdf
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Exact vs. Inexact Interpolation

• Interpolators can be either exact or inexact
• At sampled locations, exact interpolators yield
  values identical to the measurements.
• I.e., if the observed temperature in city A is 90
  degrees, the point representing city A on the
  resulting grid will still have the temperature of
  90 degrees
• At sampled locations, inexact interpolators
  predict values that are different from the
  measured values.
• I.e., if the observed temperature in city A is 90
  degrees, the inexact interpolator will still
  create a prediction for city A, and this
  prediction will not be exactly 90 degrees
• The resulting surface will not pass through the
  original point
• Can be used to avoid sharp peaks or troughs in
  the output surface
• Model quality can be assessed by the statistics
  of the differences between predicted and measured
  values
• Jumping ahead, the two deterministic
  interpolators that will be briefly presented here
  are exact. Kriging can be exact or inexact.

Reference Burrough, P. A., and R. A. McDonnell.
1998. Principles of geographical information
systems. Oxford University Press, Oxford. 333pp.
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Part 1. Deterministic Interpolation
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Inverse Distance Weighted (IDW)

• IDW interpolation explicitly relies on the First
  Law of Geography. To predict a value for any
  unmeasured location, IDW will use the measured
  values surrounding the prediction location.
  Measured values that are nearest to the
  prediction location will have greater influence
  (i.e., weight) on the predicted value at that
  unknown point than those that are farther away.
• Thus, IDW assumes that each measured point has a
  local influence that diminishes with distance (or
  distance to the power of q gt 1), and weighs the
  points closer to the prediction location greater
  than those farther away, hence the name inverse
  distance weighted.
• Inverse Squared Distance (i.e., q2) is a widely
  used interpolator
• For example, ArcGIS allows you to select the
  value of q.
• Weights of each measured point are proportional
  to the inverse distance raised to the power value
  q. As a result, as the distance increases, the
  weights decrease rapidly. How fast the weights
  decrease is dependent on the value for q.

Source http//webhelp.esri.com/arcgisdesktop/9.2/
index.cfm?TopicNameHow_Inverse_Distance_Weighted_
(IDW)_interpolation_works
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Inverse Distance Weighted - Continued

• Because things that are close to one another are
  more alike than those farther away, as the
  locations get farther away, the measured values
  will have little relationship with the value of
  the prediction location.
• To speed up the computation we might only use
  several points that are the closest
• As a result, it is common practice to limit the
  number of measured values that are used when
  predicting the unknown value for a location by
  specifying a search neighborhood. The specified
  shape of the neighborhood restricts how far and
  where to look for the measured values to be used
  in the prediction. Other neighborhood parameters
  restrict the locations that will be used within
  that shape.
• The output surface is sensitive to clustering and
  the presence of outliers.

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Search Neighborhood Specification
5 nearest neighbors with known values (shown in
red) of the unknown point (shown in
black) will be used to determine its value
Points with known values of elevation that are
outside the circle are just too far from the
target point at which the elevation value is
unknown, so their weights are pretty much 0.
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The Accuracy of the Results

• One way to assess the accuracy of the
  interpolation is known as cross-validation
• Remember the initial goal use all the measured
  points to create a surface
• However, assume we remove one of the measured
  points from our input, and re-create the surface
  using all the remaining points.
• Now, we can look at the predicted value at that
  removed point and compare it to the points
  actual value!
• We do the same thing for all the points
• If the average (squared) difference between the
  actual value and the prediction is small, then
  our model is doing a good job at predicting
  values at unknown points. If this average squared
  difference is large, then the model isnt that
  great. This average squared difference is called
  mean square error of prediction. For instance,
  the Geostatistical Analyst of ESRI reports the
  square root of this average squared difference
• Cross-validation is used in other interpolation
  methods as well

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A Cross-Validation Example

• Assume you have measurements at 15 data points,
  from which you want to create a prediction
  surface
• The Measured column tells you the measured value
  at that point. The Predicted column tells you the
  prediction at that point when we remove it from
  the input (i.e., use the other 14 points to
  create a surface). The Error column is simply the
  difference between the measured and predicted
  values.
• Because we can have an over-prediction or
  under-prediction at any point, the error can be
  positive or negative. So averaging the errors
  wont do us much good if we want to see the
  overall error well end up with a value that is
  essentially zero due to these positives and
  negatives
• Thus, in order to assess the extent of error in
  our prediction, we square each term, and then
  take the average of these squared errors. This
  average is called the mean squared error (MSE)
• For example, ArcGIS reports the square root of
  this mean squared error (referred to as simply
  Root-Mean-Square in Geostatistical Analyst). This
  root mean square error is often denoted as RMSE.

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Examples of IDW with Different qs

• Larger qs (i.e., power to which distance is
  raised) yield smoother surfaces
• Food for thought What happens when q is set to 0?

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Part 2. A Review of Stats 101
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Before we do any Geostatistics

• Lets review some basic statistical topics
• Normality
• Variance and Standard Deviations
• Covariance and Correlation
• and then briefly re-examine the underlying
  premise of most spatial statistical analyses
• Autocorrelation

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Normality

• A lot of statistical tests including many in
  geostatistics rely on the assumption that the
  data are normally distributed
• When this assumption does not hold, the results
  are often inaccurate

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Data Transformations

• Sometimes, it is possible to transform a
  variables distribution by subjecting it to some
  simple algebraic operation.
• The logarithmic transformation is the most widely
  used to achieve normality when the variable is
  positively skewed (as in the image on the left
  below)
• Analysis is then performed on the transformed
  variable.

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The Mean and the Variance
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Example Calculation of Mean and Variance
Person Test Score Distance from the Mean (Distance from the Mean) Squared
1 90 15 225
2 55 -20 400
3 100 25 625
4 55 -20 400
5 85 10 100
6 70 -5 25
7 80 5 25
8 30 -45 2025
9 95 20 400
10 90 15 225
  Mean 75   Variance 445 (Average of the entries in this column)
      Standard deviation (Square root of the variance) 21.1
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Covariance and Correlation

• Defined as a measure of how much two variables X
  and Y change together
• The units of Cov (X, Y) are those of X multiplied
  by those of Y
• The covariance of a variable X with itself is
  simply the variance of X
• Since these units are fairly obscure, a
  dimensionless measure of the strength of the
  relationship between variables is often used
  instead. This measure is known as the
  correlation.
• Correlations range from -1 to 1, with positive
  values close to one indicating a strong direct
  relationship and negative values close to -1
  indicating a strong inverse relationship

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Spatial Autocorrelation

• Sometimes, rather than examining the association
  between two variables, we might look at the
  relationship of values within a single variable
  at different time points or locations
• There is said to be (positive) autocorrelation in
  a variable if observations that are closer to
  each other in space have related values (recall
  Toblers Law)
• As an aside, there could also be temporal
  autocorrelation i.e., values of a variable at
  points close in time will be related

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Examples of Spatial Autocorrelation
(Source http//image.weather.com/images/maps/curr
ent/acttemp_720x486.jpg)
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Examples of Spatial Autocorrelation (Contd)
(Source http//capita.wustl.edu/CAPITA/CapitaRepo
rts/localPM10/gifs/elevatn.gif)
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Regression

• A statistical method used to examine the
  relationship between a variable of interest and
  one or more explanatory variables
• Strength of the relationship
• Direction of the relationship
• Often referred to as Ordinary Least Squares (OLS)
  regression
• Available in all statistical packages
• Note that the presence of a relationship does not
  imply causality

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For the purposes of demonstration, lets focus on
a simple version of this problem

• Variable of interest (dependent variable)
• E.g., education (years of schooling)
• Explanatory variable (AKA independent variable or
  predictor)
• E.g., Neighborhood Income

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But what does a regression do? An example with a
single predictor
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The example on the previous page can be easily
extended to cases when we have more than one
predictor

• When we have ngt1 predictors, rather than getting
  a line in 2 dimensions, we get a line in n1
  dimensions (the 1 accounts for the dependent
  variable)
• Each independent variable will have its own slope
  coefficient which will indicate the relationship
  of that particular predictor with the dependent
  variable, controlling for all other independent
  variables in the regression.
• The equation of the best fit line becomes
• Dep. Variable m1predictor1 m2predictor2
  m3predictor 3 b residuals
• where the ms are the coefficients of the
  corresponding predictors and b is the y-intercept
  term
• The coefficient of each predictor may be
  interpreted as the amount by which the dependent
  variable changes as the independent variable
  increases by one unit (holding all other
  variables constant)

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Some (Very) Basic Regression Diagnostics

• R-squared the percent of variance in the
  dependent variable that is explained by the
  independent variables
• The so-called p-value of the coefficient
• The probability of getting a coefficient (slope)
  value as far from zero as we observe in the case
  when the slope is actually zero
• When p is less than 0.05, the independent
  variable is considered to be a statistically
  significant predictor of the dependent variable
• One p-value per independent variable
• The sign of the coefficient of the independent
  variable (i.e., the slope of the regression line)
  
• One coefficient per independent variable
• Indicates whether the relationship between the
  dependent and independent variables is positive
  or negative
• We should look at the sign when the coefficient
  is statistically significant

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Some (but not all) regression assumptions

1. The dependent variable should be normally
   distributed (i.e., the histogram of the variable
   should look like a bell curve)
2. Very importantly, the observations should be
   independent of each other. (The same holds for
   regression residuals). If this assumption is
   violated, our coefficient estimates could be
   wrong!

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Part 3. Geostatistical Interpolation
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Origins

• Involve a set of statistical techniques called
  Kriging (there are a bunch of different Kriging
  methods)
• Kriging is named after Danie Gerhardus Krige, a
  South African mining engineer who presented the
  ideas in his masters thesis in 1951. These ideas
  were later formalized by a prominent French
  mathematician Georges Matheron
• For more information, see
• Krige, Danie G. (1951). "A statistical approach
  to some basic mine valuation problems on the
  Witwatersrand". J. of the Chem., Metal. and
  Mining Soc. of South Africa 52 (6) 119139.
• Matheron, Georges (1962). Traité de
  géostatistique appliquée, Editions Technip,
  France
• Kriging has two parts the quantification of the
  spatial structure in the data (called
  variography) and prediction of values at unknown
  points

Souce of this information http//en.wikipedia.org
/wiki/Daniel_Gerhardus_Krige
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Motivating Example Ordinary Kriging

• Imagine we have data on the concentration of gold
  (denote it by Y) in western Pennsylvania at a set
  of 200 sample locations (call them points
  p1p200).
• Since Y has a meaningful value at every point,
  our goal is to create a prediction surface for
  the entire region using these sample points
• Notation In this western PA region, Y(p) will
  denote the concentration level of gold at any
  point p.

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Global and Local Structure

• Without any a priori knowledge about the
  distribution of gold in Western PA, we have no
  theoretical reason to expect to find different
  concentrations of gold at different locations in
  that region.
• I.e., theoretically, the expected value of gold
  concentration should not vary with latitude and
  longitude
• In other words, we would expect that there is
  some general, average, value of gold
  concentration (called global structure) that is
  constant throughout the region (even though we
  assume its constant, we do not know what its
  value is)
• Of course, when we look at the data, we see that
  there is some variability in the gold
  concentrations at different points. We can
  consider this to be a local deviation from the
  overall global structure, known as the local
  structure or residual or error term.
• In other words, geostatisticians would decompose
  the value of gold Y(p) into the global structure
  µ(p) and local structure e(p).
• Y(p) µ(p) e(p)

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e(p)

• As per the First Law of Geography, the local
  structures e(p) of nearby observations will often
  be correlated. That is, there is still some
  meaningful information (i.e., spatial
  dependencies) that can be extracted from the
  spatially dependent component of the residuals.
• So, our ordinary kriging model will
• Estimate this constant but unknown global
  structure µ(p), and
• Incorporate the dependencies among the residuals
  e(p). Doing so will enable us to create a
  continuous surface of gold concentration in
  western PA.

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Assumptions of Ordinary Kriging

• For the sake of the methods that we will be
  employing, we need to make some assumptions
• Y(p) should be normally distributed
• The global structure µ(p) is constant and unknown
  (as in the gold example)
• Covariance between values of e depends only on
  distance between the points,
• To put it more formally, for each distance h and
  each pair of locations p and t within the region
  of interest that are h units are apart, there
  exists a common covariance value, C(h), such that
  covariance e(p), e(t) C(h).
• This is called isotropy

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Covariance and Distance

• From the First Law of Geography it would then
  follow that as distance between points increases,
  the similarity (i.e., covariance or correlation)
  between the values at these points decreases
• If we plot this out, with inter-point distance h
  on the x-axis, and covariance C(h) on the y-axis,
  we get a graph that looks something like the one
  below. This representation of covariance as a
  function of distance is called as the covariogram
• Alternatively, we can plot correlation against
  distance (the correlogram)

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Covariograms and Weights

• Geostatistical methods incorporate this
  covariance-distance relationship into the
  interpolation models
• More specifically, this information is used to
  calculate the weights
• As IDW, kriging is a weighted average of points
  in the vicinity
• Recall that in IDW, in order to predict the value
  at an unknown point, we assume that nearer points
  will have higher weights (i.e., weights are
  determined based on distance)
• In geostatistical techniques, we calculate the
  distances between the unknown point at which we
  want to make a prediction and the measured points
  nearby, and use the value of the covariogram for
  those distances to calculate the weight of each
  of these surrounding measured points.
• I.e., the weight of a point h units away will
  depend on the value of C(h)

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But

• Unfortunately, it so happens that one generally
  cannot estimate covariograms and correlograms
  directly
• For that purpose, a related function of distance
  (h) called the semi-variogram (or simply the
  variogram) is calculated
• The variogram is denoted by ?(h)
• One can easily obtain the covariogram from the
  variogram (but not the other way around)
• Covariograms and variograms tell us the spatial
  structure of the data

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Interpretation of Variograms

• As mentioned earlier, a covariogram might be
  thought of as covariance (i.e., similarity)
  between point values as a function of distance,
  such that C(h) is greater at smaller distances
• A variogram, on the other hand, might be thought
  of as dissimilarity between point values as a
  function of distance, such that the
  dissimilarity is greater for points that are
  farther apart
• Variograms are usually interpreted in terms of
  the corresponding covariograms or correlograms
• A common mistake when interpreting variograms is
  to say that variance increases with distance.

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Bandwidth (The Maximum Value of h)

• When there are n points, the number of
  inter-point distances is equal to
• Example
• With 15 points, we have 15(15-1)/2 105
  inter-point distances (marked in yellow on the
  grid in the lower left)
• Since were using Euclidean distance, the
  distance between points 1 and 2 is the same as
  the distance between points 2 and 1, so we count
  it only once. Also, the distance between a point
  and itself will always be zero, and is of no
  interest here.
• The maximum distance h on a covariogram or
  variogram is called the bandwidth, and should
  equal half the maximum inter-point distance.
• In the figure on the lower right, the blue line
  connects the points that are the farthest away
  from each other. The bandwidth in this example
  would then equal to half the length of the blue
  line

h
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Mathematical definition of a variogram

• In other words, for each distance h between 0 and
  the bandwidth
• Find all pairs of points i and j that are
  separated by that distance h
• For each such point pair, subtract the value of Y
  at point j from the value of Y at point i, and
  square the difference
• Average these square distances across all point
  pairs and divide the average by 2. Thats your
  variogram value!
• Division by 2 -gt hence the occasionally used name
  semi-variogram
• However, in practice, there will generally be
  only one pair of points that are exactly h units
  apart, unless were dealing with regularly spaced
  samples. Therefore, we create bins, or distance
  ranges, into which we place point pairs with
  similar distances, and estimate ? only for
  midpoints of these bins rather than at all
  individual distances.
• These bins are generally of the same size
• Its a rule of thumb to have at least 30 point
  pairs per bin
• We call these estimates of ?(h) at the bin
  midpoints the empirical variogram

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Fitting a Variogram Model

• Now, were going to fit a variogram model (i.e.,
  curve) to the empirical variogram
• That is, based on the shape of the empirical
  variogram, different variogram curves might be
  fit
• The curve fitting generally employs the method of
  least squares the same method thats used in
  regression analysis

A very comprehensive guide on variography by Dr.
Tony Smith (University of Pennsylvania)
http//www.seas.upenn.edu/ese502/NOTEBOOK/Part_II
/4_Variograms.pdf
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The Variogram Parameters

• The variogram models are a function of three
  parameters, known as the range, the sill, and the
  nugget.
• The range is typically the level of h at the
  correlation between point values is zero (i.e.,
  there is no longer any spatial autocorrelation)
• The value of ? at r is called the sill, and is
  generally denoted by s
• The variance of the sample is used as an estimate
  of the sill
• Different models have slightly different
  definitions of these parameters
• The nugget deserves a slide of its own

Graph taken from http//www.geog.ubc.ca/courses/g
eog570/talks_2001/Variogr1neu.gif
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Spatial Independence at Small Distances

• Even though we assume that values at points that
  are very near each other are correlated, points
  that are separated by very, very small values
  might be considerably less correlated
• E.g. you might find a gold nugget and no more
  gold in the vicinity
• In other words, even though ?(0) is always 0,
  however ? at very, very small distances will be
  equal to a value a that is considerably greater
  than 0.
• This value denoted by a is called the nugget
• The ratio of the nugget to the sill is known as
  the nugget effect, and may be interpreted as the
  percentage of variation in the data that is not
  spatial
• The difference between the sill and the nugget is
  known as the partial sill
• The partial sill, and not the sill itself, is
  reported in GeoStatistical Analyst

48
Pure Nugget Effect Variograms

• Pure nugget effect is when the covariance between
  point values is zero at all distances h
• That is, there is absolutely no spatial
  autocorrelation in the data (even at small
  distances)
• Pure nugget effect covariogram and variogram are
  presented below
• Interpolation wont give a reasonable predictions
• Most cases are not as extreme and have both a
  spatially dependent and a spatially independent
  component, regardless of variogram model chosen
  (discussed on following slides)

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The Spherical Model

• The spherical model is the most widely used
  variogram model
• Monotonically non-decreasing
• I.e., as h increases, the value of ?(h) does not
  decrease - i.e., it goes up (until hr) or stays
  the same (hgtr)
• ?(hr)s and C(hr)0
• That is, covariance is assumed to be exactly zero
  at distances hr

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The Exponential Model

• The exponential variogram looks very similar to
  the spherical model, but assumes that the
  correlation never reaches exactly zero,
  regardless of how great the distances between
  points are
• In other words, the variogram approaches the
  value of the sill asymptotically
• Because the sill is never actually reached, the
  range is generally considered to be the smallest
  distance after which the covariance is 5 or less
  of the maximum covariance
• The model is monotonically increasing
• I.e., as h goes up, so does ?(h)

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The Wave (AKA Hole-Effect) Model
On the picture to the left, the waves exhibit a
periodic pattern. A non-standard form of spatial
autocorrelation applies. Peaks are similar in
values to other peaks, and troughs are similar in
values to other troughs. However, note the
dampening in the covariogram and variogram below
That is, peaks that are closer together have
values that are more correlated than peaks that
are father apart (and same holds for troughs).
More is said about the applicability of these
models in ttp//www.gaa.org.au/pdf/gaa_pyrcz_deuts
ch.pdf Variogram graph edited slightly from
http//www.seas.upenn.edu/ese502/NOTEBOOK/Part_II
/4_Variograms.pdf
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Variograms and Kriging Weights
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Reviewing Ordinary Kriging

• Again, ordinary kriging will
• Give us an estimate of the constant but unknown
  global structure µ(p), and
• Use variography to examine the dependencies among
  the residuals e(p) and to create kriging weights.
• We calculate the distances between the unknown
  point at which we want to make a prediction and
  the measured points that are nearby and use the
  value of the covariogram for those distances to
  calculate the weight of each of these surrounding
  measured points.
• The end result is, of course, a continuous
  prediction surface
• Prediction standard errors can also be obtained
  this is a surface indicating the accuracy of
  prediction

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Universal Kriging

• Now, take another example imagine we have data
  on the temperature at 100 different weather
  stations (call them w1..w100) throughout Florida,
  and we want to predict the values of temperature
  (T) at every point w in the entire state using
  these data.
• Notation temperature at point w is denoted by
  T(w)
• We know that temperature at lower latitudes are
  expected to be higher. So, T(w) will be expected
  to vary with latitude
• Ordinary kriging is not appropriate here, because
  it assumes that the global structure is the same
  everywhere. This is clearly not the case here.
• A method called universal kriging allows for a
  non-constant global structure
• We might model the global structure µ as in
  regression
• Everything else in universal kriging is pretty
  much the same as in ordinary kriging (e.g.,
  variography)

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Some More Advanced Techniques

• Indicator Kriging is a geostatistical
  interpolation method does not require the data to
  be normally distributed.
• Co-kriging is an interpolation technique that is
  used when there is a second variable that is
  strongly correlated with the variable from which
  were trying to create a surface, and which is
  sampled at the same set of locations as our
  variable of interest and at a number of
  additional locations.
• For more details on indicator kriging and
  co-kriging, see one of the texts suggested at the
  end of this presentation

56
Isotropy vs. Anisotropy

• When we use isotropic (or omnidirectional)
  covariograms, we assume that the covariance
  between the point values depends only on distance
• Recall the covariance stationarity assumption
• Anisotropic (or directional) covariograms are
  used when we have reason to believe that
  direction plays a role as well (i.e., covariance
  is a function of both distance and direction)
• E.g., in some problems, accounting for direction
  is appropriate (e.g., when wind or water currents
  might be a factor)

For more on anisotropic variograms, see
http//web.as.uky.edu/statistics/users/yzhen8/STA6
95/lec05.pdf
57
IDW vs. Kriging

• We get a more natural look to the data with
  Kriging
• You see the bulls eye effect in IDW but not (as
  much) in Kriging
• Helps to compensate for the effects of data
  clustering, assigning individual points within a
  cluster less weight than isolated data points
  (or, treating clusters more like single points)
• Kriging also give us a standard error
• If the data locations are quite dense and
  uniformly distributed throughout the area of
  interest, we will get decent estimates regardless
  of which interpolation method we choose.
• On the other hand, if the data locations fall in
  a few clusters and there are gaps in between
  these clusters, we will obtain pretty unreliable
  estimates regardless of whether we use IDW or
  Kriging.

These are interpolation results using the gold
data in Western PA (IDW vs. Ordinary Kriging)
58
Some Widely Used Texts on Geostatistics

• Bailey, T.C. and Gatrell, A.C. (1995) Interactive
  Spatial Data Analysis. Addison Wesley Longman,
  Harlow, Essex.
• Cressie, N.A.C. (1993) Statistics for Spatial
  Data. (Revised Edition). Wiley, John Sons,
  Inc.,
• Isaaks, E.H. and Srivastava, R.M. (1989) An
  Introduction to Applied Geostatistics. Oxford
  University Press, New York, 561 p.